2.2. LAPLACE’S EQUATION 29

c. Local estimates for harmonic functions. Now we employ the mean-

value formulas to derive careful estimates on the various partial derivatives

of a harmonic function. The precise structure of these estimates will be

needed below, when we prove analyticity.

THEOREM 7 (Estimates on derivatives). Assume u is harmonic in U.

Then

(18)

|Dαu(x0)|

≤

Ck

rn+k

u

L1(B(x0,r))

for each ball B(x0,r) ⊂ U and each multiindex α of order |α| = k.

Here

(19) C0 =

1

α(n)

, Ck =

(2n+1nk)k

α(n)

(k = 1,... ).

Proof. 1. We establish (18), (19) by induction on k, the case k = 0 being

immediate from the mean-value formula (16). For k = 1, we note upon

diﬀerentiating Laplace’s equation that uxi (i = 1,...,n) is harmonic. Con-

sequently

(20)

|uxi (x0)| = −

B(x0,r/2)

uxi dx

=

2n

α(n)rn

∂B(x0,r/2)

uνi dS

≤

2n

r

u

L∞(∂B(x0,

r

2

)).

Now if x ∈

∂B(x0,r/2), then B(x, r/2) ⊂ B(x0,r) ⊂ U, and so

|u(x)| ≤

1

α(n)

2

r

n

u

L1(B(x0,r))

by (18), (19) for k = 0. Combining the inequalities above, we deduce

|Dαu(x0)|

≤

2n+1n

α(n)

1

rn+1

u

L1(B(x0,r))

if |α| = 1. This veriﬁes (18), (19) for k = 1.

2. Assume now k ≥ 2 and (18), (19) are valid for all balls in U and each

multiindex of order less than or equal to k − 1. Fix B(x0,r) ⊂ U and let α